課程名稱︰微積分甲上
課程性質︰數學系大一必帶
課程教師︰陳榮凱
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013/11/14
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
The total is 110 points.
(1) (30 pts) (Differential Calculus)
ｄ -1
(a) (5 pts) Determine ─tanh x.
dx
3
df(x) x + x u
(b) (5 pts) Determine ─── with f(x) = ∫ e du.
dx 0
-1
─
2
x
(c) (10 pts) Prove that f(x) = e has derivative and second derivative 0
at x = 0.
2
3x
(d) (10 pts) Consider the function f(x) = ──. Sketch the graph of f(x) by
2
x -1
locating all relative maxima, minima, inflection points, non-
differentiable points if any. Also determine lim f(x) and lim f(x).
x→∞ x→-∞
(2) (30 pts) (Integral Calculus)
π/2
(a) (5 pts) Evaluate ∫ sec x tan x dx.
π/3
4x
(b) (10 pts) Determine ∫────── dx.
2 4
(x -1)(x +1)
1 n-1 m-1 (n-1)!(m-1)!
(c) (10 pts) Show that ∫ x (1-x) dx = ────── for any positive
0 (n+m-1)!
integer n, m. This is the so-called β function.
∞ １
(d) (5 pts) Determine the convergence of ∫ ───── dx.
2 3
x(log x)
(3) (20 pts) (Continuity)
2
(a) (10 pts) Show that f(x) = x + 1 is not uniformly continuous on R.
(b) (10 pts) Prove that f(x) is continuous at x if and only if lim f(x )
0 x →x n
n 0
= f(x ) for any sequence {x } converging to x .
0 n 0
(4) (10 pts) For any a＜b＜c＜d, prove that there exists a differentiable
function f(x) on R such that f(x) = 1 if xε[b,c] and f(x) = 0 if x≧d or
x≦a.
(5) (10 pts) Let f(x) be a differentiable function on R. Suppose that f(0) = 0
and |f'(x)|≦|f(x)|. Prove that f(x) = 0 identically.
(6) (10 pts) For any n≧2εN, let v(n) be the number of prime factors of n. For
v(n)
example, v(4) = v(5) = 1, v(6) = 2, v(30) = 3. Prove that lim ── = 0.
n→∞ ｎ
//ε是屬於符號
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